The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.150.2. Let $(R, \mathfrak m, \kappa )$ be a local ring. Let $\kappa \subset \kappa ^{sep}$ be a separable algebraic closure. There exists a commutative diagram

\[ \xymatrix{ \kappa \ar[r] & \kappa \ar[r] & \kappa ^{sep} \\ R \ar[r] \ar[u] & R^ h \ar[r] \ar[u] & R^{sh} \ar[u] } \]

with the following properties

  1. the map $R^ h \to R^{sh}$ is local

  2. $R^{sh}$ is strictly henselian,

  3. $R^{sh}$ is a filtered colimit of ├ętale $R$-algebras,

  4. $\mathfrak m R^{sh}$ is the maximal ideal of $R^{sh}$, and

  5. $\kappa ^{sep} = R^{sh}/\mathfrak m R^{sh}$.

Proof. This is proved by exactly the same proof as used for Lemma 10.150.1. The only difference is that, instead of pairs, one uses triples $(S, \mathfrak q, \alpha )$ where $R \to S$ ├ętale, $\mathfrak q$ is a prime of $S$ lying over $\mathfrak m$, and $\alpha : \kappa (\mathfrak q) \to \kappa ^{sep}$ is an embedding of extensions of $\kappa $. $\square$


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